--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic_2/grammar/lenv_weight.ma".
+include "Basic_2/grammar/lsubs.ma".
+include "Basic_2/substitution/lift.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Basic_1: includes: ldrop_skip_bind *)
+inductive ldrop: nat → nat → relation lenv ≝
+| ldrop_atom: ∀d,e. ldrop d e (⋆) (⋆)
+| ldrop_pair: ∀L,I,V. ldrop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
+| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. 𝕓{I} V) L2
+| ldrop_skip: ∀L1,L2,I,V1,V2,d,e.
+ ldrop d e L1 L2 → ↑[d,e] V2 ≡ V1 →
+ ldrop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
+.
+
+interpretation "ldropping" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+#d #e #L1 #L2 * -d e L1 L2
+[ //
+| //
+| #L1 #L2 #I #V #e #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_refl *)
+lemma ldrop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5/ qed.
+
+fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
+ L2 = ⋆.
+#d #e #L1 #L2 * -d e L1 L2
+[ //
+| #L #I #V #H destruct
+| #L1 #L2 #I #V #e #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_sort *)
+lemma ldrop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
+/2 width=5/ qed.
+
+fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K. 𝕓{I} V →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[d, e - 1] K ≡ L2).
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #K #I #V #H destruct
+| #L #I #V #_ #K #J #W #HX destruct -L I V /3/
+| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
+]
+qed.
+
+lemma ldrop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[0, e - 1] K ≡ L2).
+/2/ qed.
+
+(* Basic_1: was: ldrop_gen_ldrop *)
+lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
+ ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
+#e #K #I #V #L2 #H #He
+elim (ldrop_inv_O1 … H) -H * // #H destruct -e;
+elim (lt_refl_false … He)
+qed.
+
+fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
+ ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L2 = K2. 𝕓{I} V2.
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #I #K #V #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
+ /2 width=5/
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_skip_l *)
+lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L2 = K2. 𝕓{I} V2.
+/2/ qed.
+
+fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+ ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #I #K #V #H destruct
+| #L #I #V #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
+ /2 width=5/
+]
+qed.
+
+(* Basic_1: was: ldrop_gen_skip_r *)
+lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
+ ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+/2/ qed.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was by definition: ldrop_refl *)
+lemma ldrop_refl: ∀L. ↓[0, 0] L ≡ L.
+#L elim L -L //
+qed.
+
+lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
+ ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
+#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
+qed.
+
+lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
+ ∀K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{Abbr} V →
+ d ≤ i → i < d + e →
+ ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
+ ↓[0, i] L2 ≡ K2. 𝕓{Abbr} V.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e
+[ #d #e #K1 #V #i #H
+ lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L1 #L2 #K1 #V #i #_ #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
+ elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
+ [ -IHL12 Hie; destruct -i K1 W;
+ <minus_n_O <minus_plus_m_m /2/
+ | -HL12;
+ elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
+ ]
+| #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
+ elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
+ [ -IHL12 Hie Hi; destruct
+ | elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
+ ]
+| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
+ lapply (plus_S_le_to_pos … Hdi) #Hi
+ lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1
+ elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
+]
+qed.
+
+(* Basic forvard lemmas *****************************************************)
+
+(* Basic_1: was: ldrop_S *)
+lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
+ ↓[O, e + 1] L1 ≡ K2.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e K2 I2 V2 /2/
+ | @ldrop_ldrop >(plus_minus_m_m e 1) /2/
+ ]
+]
+qed.
+
+lemma ldrop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
+#L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize
+[ /2/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(tw_lift … HV21) -HV21 /2/
+]
+qed.
+
+lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
+ ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e K2 I2 V2 //
+ | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
+ ]
+]
+qed.
+
+lemma ldrop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 elim L1 -L1
+[ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
+| #K1 #I1 #V1 #IHL1 #L2 #e #H
+ elim (ldrop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e L2 //
+ | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
+ >minus_le_minus_minus_comm //
+ ]
+]
+qed.
+
+(* Basic_1: removed theorems 49:
+ ldrop_skip_flat
+ cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+ ldrop_clear ldrop_clear_O ldrop_clear_S
+ clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+ clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+ getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+ getl_clear_bind getl_clear_conf getl_dec getl_ldrop getl_ldrop_conf_lt
+ getl_ldrop_conf_ge getl_conf_ge_ldrop getl_ldrop_conf_rev
+ ldrop_getl_trans_lt ldrop_getl_trans_le ldrop_getl_trans_ge
+ getl_ldrop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+ getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)
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